Fold and thrust partitioning in a contracting fold belt: Insights

from the 1931 Mach earthquake in Baluchistan

Walter Szeliga,1 Roger Bilham,1 Daniel Schelling,2 DinMohamedKakar,3 and Sarosh Lodi4

Received 21 January 2008; revised 27 October 2008; accepted 8 July 2009; published 30 October 2009.

[1] Surface deformation associated with the 27 August1931 earthquake near Mach in Baluchistan is quantifiedfrom spirit-leveling data and from detailed structuralsections of the region interpreted from seismic reflectiondata constrained by numerous well logs. Mean slip onthewest dippingDezghat/Bannh fault system amountedto 1.2 m on a 42 km � 72 km thrust plane with sliplocally attaining 3.2 m up dip of an inferred locking lineat �9 km depth. Slip also occurred at depths below theinterseismic locking line. In contrast, negligible slipoccurred in the 4 km near the interseismic locking line.The absence of slip here in the 4 years following theearthquake suggests that elastic energy there must eitherdissipate slowly in the interseismic cycle, or that a slipdeficit remains, pending its release in a large futureearthquake. Elastic models of the earthquake cycle inthis fold and thrust belt suggest that slip on the frontalthrust fault is reduced by a factor of 2 to 8 compared tothat anticipated from convergence of the hinterland, apartitioning process that is presumably responsible forthickening of the fold and thrust belt at the expenseof slip on the frontal thrust. Near the latitude ofQuetta, GPSmeasurements indicate that convergence is�5 mm/yr. Hence the minimum renewal time betweenearthquakes with 1.2-m mean displacement should beas little as 240 years. However, when the partitioningof fold belt convergence to frontal thrust slip is takeninto account the minimum renewal time may exceed2000 years. Citation: Szeliga, W., R. Bilham, D. Schelling,

D. M. Kakar, and S. Lodi (2009), Fold and thrust partitioning in a

contracting fold belt: Insights from the 1931 Mach earthquake in

Baluchistan, Tectonics, 28, TC5019, doi:10.1029/2008TC002265.

1. Introduction

[2] Between 1931 and 1935 three major earthquakesoccurred between the Bolan Pass and Quetta in the Baluchi-stan province of Pakistan. The first of these, an Mw 6.8 event

near Sharigh (2135 UT, 24 August 1931), was followed 66 hlater by the Mach Mw 7.3 earthquake (1527 UT, 27 August).The third and largest earthquake was the Mw 7.7 30 May1935 earthquake that destroyed 90% of Quetta and caused35,000 deaths [Ambraseys and Bilham, 2003]. Because nosimilar magnitude earthquakes occurred in the 3 decadesbefore or after this sequence, it is very probable that staticor dynamic triggering of these nearby earthquakes is respon-sible for their clustering in time.[3] All three earthquakes lie within the 150-km-wide zone

of deformation between the Asian and Indian plates, a regionbounded to the west by the Chaman fault and to the east bythe Indus plain [Bender and Raza, 1995] (Figure 1a). Thestrike-slip component of slip on the plate boundary isestimated to be 33 cm/yr from global GPS closure estimates[Apel et al., 2006] and 31 mm/yr from paleomagneticreconstructions of the Indian Ocean seafloor [Molnar andStock, 2009]. Geological estimates of slip on the Chamanfault system [Lawrence et al., 1992] indicate slip rates of 19–24 mm/yr in the past 20 Myr and 25–35 mm/yr for the past2 Myr [Beun et al., 1979; Lawrence et al., 1992], suggestingthat as much as one third of this shear signal may bedistributed in the fold belt. The plate boundary is regionallytranspressive and from seismic moment release calculationsin the past 200 years, and from the inferred obliquity of theplate boundary to the local slip vector between India and Asiait has been estimated that strain partitioning results in con-vergence of the fold belts of up to 13 ± 3 mm/yr [Ambraseysand Bilham, 2003]. This estimate is probably inflated by theseismic productivity of the past century, which may beabnormally high if the earthquakes presently under discus-sion are atypical of long-term seismicity. A lower conver-gence rate is obtained from analog and numerical modelingof the strain in the region (3–6 mm/yr NW/SE shortening[Bernard et al., 2000; Haq and Davis, 1997]), and this lowerrate is consistent with preliminary GPS measurements northof Quetta presented in this paper.[4] Although triangulation data exist in the region no

remeasurements have been published [Ambraseys and Bilham,2003]. However, a first-order spirit leveling line, first mea-sured in 1909 between Sukkur and Chaman, was remeasuredshortly after the 1935 Quetta earthquake [Wilson, 1938].Parts of the line were raised 65 cm where they crossed thefrontal thrusts of the northern Kirthar Range to the west ofSibi, exceeding the combined errors in the survey by morethan an order of magnitude. A preliminary analysis of thesedata (Figure 2a) in the absence of geological constraints,concluded that the asymmetry in the vertical deformation,if caused by planar slip, was caused by 1–1.2 m of slip onan east dipping blind thrust fault between 1 km depth and

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1CIRES and Department of Geological Sciences, University of Colorado,Boulder, Colorado, USA.

2Structural Geology International, LLC, Salt Lake City, Utah, USA.3Department of Geology, University of Baluchistan, Quetta, Pakistan.4Department of Civil Engineering, NED University, Karachi, Pakistan.

Copyright 2009 by the American Geophysical Union.0278-7407/09/2008TC002265$12.00

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�25 km depth [Ambraseys and Bilham, 2003]. The fitbetween observed surface deformation and synthetic planarslip was appealing, but the asymmetry of the uplift signalrequired a counterintuitive easterly dip to the frontal thrust.The description of a wedge-thrust geometry with an east

dipping shallow ramp at this location in the literature, how-ever, appeared to confirm its presence [Banks and Warburton,1986]. Yet, when the surface deformation associated with slipon the two faults of their triangle zone was examined in detailbyGarcia et al. [2006], no combination of slip was found that

Figure 1. (a) Recent seismicity (Mw > 5) and instrumental locations for the Sharigh, Mach (stars), andQuetta earthquakes (focal mechanism beach ball) and their inferred causal faults (Quetta rupture dashed andBannh fault shown as surface thrust NE of the instrumental epicenter). Focal mechanisms are scaledaccording to magnitude: the largest focal mechanism is Mw 7.7 [from Singh and Gupta, 1980] and thesmallest is Mw = 5 (all from the Harvard MCT). (b) Interpolated intensity VIII isoseismals for the threeearthquakes, the path of the 1909–1936 leveling line, and GPS velocity vectors 2005–2008 relative tofixed India. The approximate rupture zone of theMach earthquake is shown by the rectangle. The intensity-derived epicenters are shown on each map as a star. The Quetta centroid solution lies at the opposite end ofthe rupture from the intensity solution as a result of directivity.

Figure 2. Schematic sections of vertical deformation and subsurface geometry of previous attempts toemulate observed uplift data in the Mach earthquake (Figures 2a and 2b). These models invoked uniformsubsurface slip on shallow east dipping planar thrusts. (a) Planar, uniform slip is invoked with no structuralcontrol [Ambraseys and Bilham, 2003]. (b) The speculative wedge thrust geometry ofBanks andWarburton[1986] constrains two fault planes on which combinations of uniform slip were imposed to obtain thebest fitting surface uplift [Garcia et al., 2006]. (c) Spatially variable slip on the west dipping Bannh fault[Bannert et al., 1992; Schelling, 1999] is investigated in this paper.

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resulted in an improved fit to the leveling data (Figure 2b).Models with more complex geometries also failed to improvethe fit.[5] The two previous interpretations, based as they were

on limited structural information may now be discarded dueto the availability of seismic reflection data from the Sibi andMach areas, controlled by stratigraphic information fromnumerous boreholes. We present structural interpretationsin section 2 that demonstrate that the frontal thrusts of theKirthar range west of Sibi dip to the west [Bannert et al.,1992; Schelling, 1999].

2. Structural Setting of the Bolan Pass Region

[6] The Bolan Pass region of Baluchistan (Figure 3) islocated along the deformation front of the northern KirtharRange, where north-south trending structural systems (foldsand thrust faults) of the Kirthar Range give way to the morecomplexly oriented structural systems of the Quetta Syntaxis.Detailed structural field work carried out in the Bolan Passregion and elsewhere in the Kirthar Range during the late1990s [Schelling, 1999] indicates that the deformation frontof the northern Kirthar Range is dominated by east vergent,contractional fold-fault systems that give way to strike-sliporiented fault systems along and to the west of the QuettaPlateau. In addition, surface structural data has allowed thegeometries and orientations of fold-fault pairs to be definedacross and along the mountain front at different tectonostrati-graphic levels in the Bolan Pass region. Surface structuralgeometries from the Bolan Pass have been projected to depth,and in conjunction with interpreted seismic data from theBolan Pass and adjacent Sibi Trough areas, a balancedstructural cross section has been constructed across the BolanPass in the vicinity of the leveling line examined in this paper(Figure 4).[7] As shown on the cross section of Figure 4, the frontal

structural system of the Bolan Pass area is defined by an eastvergent, asymmetric fault propagation fold (the DezghatAnticline) and underlying, west dipping thrust fault systemthat is known from seismic data to flatten near the base of theSiwalik Group, a roughly 6-km-thick stratigraphic section ofMiocene-Pliocene sandstone, shale, and conglomerate thatdefine foreland basin fill to the actively subsiding Sibi Trough(Indus Basin). Tectonic shortening across the Dezghat Anti-cline and associated thrust faults is on the order of severalkilometers (Figure 4). However, all of the structural upliftsidentified to the west of the Dezghat Anticline are knownfrom surface and subsurface structural data to involve theJurassic Chiltan Limestone and overlying Cretaceous throughEocene stratigraphic section, including the Goru and Sembarformations, the Parh and Dunghan limestones, the GhazijShale, and the Kirthar Limestone (Figure 4). Between theParhi Jhal Anticline mapped to the west of the Bolan Pass andthe Sibi Trough located to the east of the Kirthar Range,structural uplift of the Chiltan Limestone is on the order of9 to 10 km (Figure 4), and associated tectonic shorteningacross the same structural systems, as determined from sur-face and subsurface data, is estimated at�15 km. This requiresthat thrust faults exposed at the surface and that involve the

Chiltan Limestone are relatively high angle structural fea-tures, with measured, near-surface dips of roughly 60� andestimated fault angles of 30� to 45� at 10 to 20 km depth(Figure 4). In addition, 9 to 10 km of uplift across the com-bined Bolan Pass and Parhi Jhal Anticlines requires a mid-crustal decollement surface at a depth of 18 to 20 km beneaththe Quetta Plateau, as indicated on the cross section ofFigure 4. This 18- to-20-km decollement depth is well belowthe projected, base-Triassic stratigraphic level of knownlithology from the Kirthar and nearby Sulaiman mountainranges, and therefore, the lithology at the basal decollementlevel beneath the Quetta Plateau remains unknown and mayactually be located in basement rocks.[8] Additional decollement surfaces of the Kirthar Range

have been identified within the Eocene Ghazij Shale, whichseparates the underlying, competent carbonates of the Chiltan-Dunghan limestones from the overlying Kirthar Limestoneand Siwalik Group (Figure 4). Significant deformation asso-ciated with these Ghazij Shale decollement surfaces isrestricted to an area above the subsurface, frontal fault rampidentified beneath the Bolan Pass, where the underlying(basal) decollement surface to the Kirthar Range climbs froma depth of 18 km or more beneath the Parhi Jhal and BolanPass Anticlines to the upper, basal Siwalik Group decolle-ment surface identified beneath the Dezghat Anticline atroughly 6 km depth. Thrust faults originating within theGhazij Shale result in short (several hundred meters) wave-length anticline-syncline pairs and the development of ex-posed back thrust surfaces along the east limb of the BolanRiver Anticline. These latter fold-fault systems have accom-modated less than 1 km of tectonic shortening, though asexposed structural systems in the Bolan Pass area there islittle question that thrust faults originating within the GhazijShale will affect surface deformation across the Bolan Passarea, as indicated from the leveling data discussed in thispaper.

3. GPS Measurements of Convergence and

Shear Between the Asian and Indian Plates

[9] GPS measurements in Pakistan are historically oflimited coverage and duration. Campaign measurementsfrom six sites with locations between the town of Chaman,30 km west of the Chaman fault, and the town of Kach,�70 km NE of Quetta, have been measured at least twice inthe period 2006–2008 and compared to continuous measure-ments made in Karachi (not shown) and Quetta (qtag)(Figure 1b). The continuous points in Pakistan are operatedfrom flat-roofed concrete frame buildings and the campaignpoints are measured on bipods set on stainless steel screwscemented into exposed rock. GPS observations were recordedeither with Trimble NetRS, 5700 or R7 receivers using a 30 ssampling rate, and processed using an elevation cutoff angleof 10�. Campaign data have durations of 3–7 days from eachsite. The daily data from these sites were processed alongwith data from 10 regional IGS stations using GAMITversion 10.34 [King and Bock, 2002]. The regional solutionswere then combined with global solutions from SOPAC(ftp://garner.ucsd.edu/pub/hfiles) using GLOBK/GLORG

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Figure 3. Geological map of the Bolan Pass region of the northern Kirthar Range, showing the locationsof the balanced structural cross section and leveling line discussed in the paper.

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version 5.16 [Herring, 2002] to determine time series andvelocities in the ITRF2005 reference frame. These velocitieswere then transformed into an Indian plate-fixed referenceframe using pole of rotation parameters determined byBettinelli et al. [2006].[10] The processed campaign GPS data are associated with

formal uncertainties of ±3 mm/yr, and the continuous data areassociated with uncertainties of ±1 mm/yr (Figure 5). Notshown Figure 5 are the velocity vectors for Karachi andNagar Parkar (north of the 2001 Bhuj earthquake) that moveat approximately the velocity of the Indian plate suggestingthat little deformation occurs across the Indus delta, or nearthe Bhuj region. The SSW velocity of Quetta is anomalousrelative to the points on the east-west transect throughChaman. It is possible that the proximity of the city to thefault that slipped in 1935 may influence the azimuth of itsvector, and we have accordingly established additional pointsto the southeast of Quetta to search for local deformation;however, these data are currently too noisy to be of value tothe present study. InSAR imagery shows no evidence for aregion of subsidence near Quetta caused by groundwaterwithdrawal (S. P. Satyabala, personal communication, 2008).The most easterly point on the traverse, Kach, is alsoanomalous in that it shows no convergence with India. Wenote that Kach lies close to the Harnai fault at the confluenceof north striking and west striking structures and its velocityvector may thus be influenced by local tectonics.[11] Thus, our limited view ofmotions within this complex

region of shear and convergence afforded by the GPS datapermit only the simplest of interpretations at present. InFigure 5 we show the form of the velocity field anticipatedfrom simple shear at depth. The curve shown is thatcorresponding to 31 mm/yr of left lateral slip on the Chamanfault below 12 km depth, a locking depth consistent withthe depths of seismicity observed near the Chaman fault[Ambraseys and Bilham, 2003], but the uncertainties in thedata permit both shallower or deeper locking depths and amuch reduced slip velocity. That four of the campaign datapoints are systematically misfit by this simple view of shear

Figure 4. Balanced structural cross section across the deformation front of the northern Kirthar Range inthe Bolan Pass area and in the vicinity of the leveling line. See Figure 3 for cross section location and text fordiscussion.

Figure 5. (a) GPS velocities projected east-west showingsouthward velocities relative to stable India. For locations,see Figure 1b. The curve indicates the elastic velocity fieldcalculated for a shear dislocation on the Chaman fault slip-ping at 31 mm/yr below a locked depth of 12 km. (b) GPSvelocities projected east-west showing westward velocitiesrelative to stable India. For locations, see Figure 1b. The GPSpoints, with two exceptions, show convergence with fixedIndia at 5 ± 1 mm/yr. The two exceptions (see text) are atKach and Qtag (the continuous GPS point at Quetta).

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suggests that the surface structures are moving relative to thebasement as anticipated by structural models of thin skinnedtectonics in the region [Bernard et al., 2000; Haq and Davis,1997].[12] We interpret the mean westward translation of five of

the seven GPS points depicted in Figures 1b and 5 toward theIndian plate at 5 ± 1mm/yr as indicative of the transpressionalconvergence of the Sulaiman fold belt, which we use as aproxy for maximum convergence rates in the Kirthar range.The GPS measurements were obtained north of Quetta at�30�N, where the fold belt is significantly wider than at thelatitude of the Mach earthquake. The lower transpressionalobliquity of the Chaman system south of 30�N, and itsnarrower width, suggests that convergence occurs there at alower rate. We cannot as yet quantify this from directobservation but we assume that the rate is at least half thatof the rate measured north of Quetta, i.e., current east-westconvergence near the epicenters of the Quetta and Machearthquakes is probably 2.5 to 5 mm/yr.

4. Macroseismic Location of the Mach

Earthquake

[13] The Mach earthquake is named for the railway head-quarters at Mach that were heavily damaged in the earth-quake [Ambraseys and Bilham, 2003]. The jail wasdestroyed, and 400 prisoners were briefly at large. Althoughno surface rupture was recorded, the parapets of a 140-m-long approximately E-W bridge converged 20 cm withoutbeing tilted. Numerous rockfalls occurred at the time of theearthquake raising clouds of dust near Mach and the BolanPass to the SE.[14] Although instrumental locations and magnitudes are

available for the 1931 and 1935 earthquakes (Figure 1 andTable 1), additional knowledge of the extent and azimuth oftheir rupture zones can be inferred from intensity datarecorded for each event. The data are available in the formof damage reports to structures near their epicenters, and fromfelt reports at larger distances. Previous analysis of theintensity data for the Sharigh, Mach and Quetta earthquakeshave interpolated isoseismal contours for a range of intensi-ties to determine the most probable location of their rupturezones. Banana-shaped isoseismals drawn byWest [1934] forthe highest intensities are distorted by the uneven coverage ofhis data. Ambraseys and Bilham [2003] reevaluated thesedata supplemented by additional observations and concludedthat the highest isoseismals for the second two earthquakesform north elongated polygons (Figure 1b). Insufficient datafor the Sharigh earthquake were available to form definitiveconclusions, except that its epicentral location was close tothe town of that name.

[15] We have subjected these same observed intensitiesto a more rigorous analysis using the methods of Bakun andWentworth [1997]. This approach does not contour iso-seismals but instead contours the most probable locationsfor the epicenter using a grid search and assumptions aboutattenuation of shaking intensity with distance. The methoduses recent earthquakes from elsewhere in the region forwhich location and magnitude are known and for whichintensity data are also available, to quantify the attenuation ofshaking intensity with distance [Szeliga et al., 2009]. Themethod then calculates the most likely magnitude for theearthquake, were it to have occurred at points on a hypothet-ical regularly spaced grid centered on the centroid of maxi-mum intensities. The resulting values on the grid are thencontoured to provide a series of isomagnitude contourssurrounding a closed contour of minimum magnitude forthe earthquake. The center of this minimum contour is theepicentral location of the smallest possible earthquake thatcould have caused the observed intensity distribution(Figure 6 and Table 1). If observations are noise free, welldistributed in intensity range, and endowed with goodazimuthal coverage, these contours tend to be elliptical orcircular; however, if the coverage is azimuthally poor, or oflow quality, the resulting contours may be complex withmultiple minima.[16] The magnitude contours provide no estimate of the

variance between magnitudes predicted from each observa-tion at each point in the grid search. A new set of contours isgenerated based on the variance of the magnitude estimatesderived for each point on the grid were the earthquake to haveoccurred at that point. When these variances are contoured, aregion of minimum variance is obtained, usually, but notalways, close to the minimum magnitude solution. For thethree earthquakes in Baluchistan the minimum magnitudelocations are indeed close to their minimum variance loca-tions, typical of an acceptablemacroseismic solution (Table 1).The minimum variance contours are assigned probabilities,with the most probable location for the epicenter beingthe location of minimum variance. The intersection of theminimum variance location with a contour of the isomagni-tude solution indicates the most probable magnitude for theearthquake.[17] The solutions for all three earthquakes are listed in

Table 1 and are shown in Figure 6. The absence of complexityto the minimum magnitude contours and their coincidencewith the marginally more complex minimum variance loca-tions in each case is notable. The most probable location forthe Mach epicenter lies near the southern end of its rupturezone and near its instrumental location.We ignore the smallerminimum noted north of Sibi. In contrast the location of theminimum variance epicenter for the 1935 Quetta earthquakelies near Quetta, more than 130 km NNE of the instrumental

Table 1. Instrumental and Inferred Macroseismic Locations for the Three Earthquakes

Event Date Instrumental Epicenter Mw Minimum Magnitude Epicenter Mi Minimum Variance Epicenter

Sharigh 24 Aug 1931 31.1�N, 67.7�E 6.8 29.87�N, 67.62�E 5.9 30.12�N, 67.60�EMach 27 Aug 1931 29.9�N, 67.6�E 7.3 29.55�N, 67.55�E 7.2 29.22�N, 67.47�EQuetta 30 May 1935 28.87�N, 66.4�E 7.7 30.10�N, 66.92�E 7.6 30.18�N, 66.92�E

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location for the earthquake. We assume that the high inten-sities reported from Quetta were enhanced by directivity inthe direction of rupture propagation [Singh and Gupta, 1980;Day et al., 2008]. The minimum variance epicentral locationfor the Sharigh event is found to be approximately to thenorth of the Mach event, west of the epicentral location in-ferred by earlier investigations; however, similar to previousinvestigations of this earthquake, we are unable to identify aspecific causal fault.[18] Using the constants derived by Bakun and Wentworth

[1997] for California, we infer the Mach magnitude to havebeen Mw 7.2 in good agreement with the value of Mw 7.3derived by Ambraseys and Bilham [2003], but the inferredmagnitude for the Sharigh event is significantly smaller,Mw 5.9 instead of 6.8, probably the result of the sparsesampling of macroseismic data points for this event. Use ofconstants derived by Szeliga et al. [2009] yield similarepicentral locations but with lower magnitudes for all threeearthquakes. Our preferred location for the Sharigh event lies

to the west of previously inferred locations, but no causalfault can be identified there from geological evidence ormicroseismicity. Recent CMT solutions [Giardini et al.,1985;Dziewonski et al., 1999] in the area (Figure 1a) indicatethat the newly located Sharigh event lies in a transitionregion between NW-SE directed compression in the BolanPass region and N-S compression to the north. Focal mech-anisms nearest to the epicenter of the Sharigh earthquakeshow a combination of NE dipping thrust faulting andthrusting along N-S directed decollements.

5. Leveling Data

[19] The Sukkur-Quetta leveling line (Figures 1b and 7)was originally surveyed between 1909 and 1914 with abenchmark spacing of 2–5 km [Wilson, 1938; Ambraseysand Bilham, 2003]. The leveling data are associated withrandom errors that grow with the square root of the distancetraversed (L km) as k

pL mm, where k = 0.65, a constant

Figure 6. Macroseismc epicenters for the Sharigh, Mach, and Quetta earthquakes. The dashed contoursare not isoseismals but isomagnitude contours using the method of Bakun and Wentworth [1997]. Theyindicate the required magnitude for each earthquake had it been located on these contours. The preferredmacroseismic epicentral location lies within the closed contour of the minimum-variance solution shown assolid lines, while the stars represent the instrumentally located epicenters.

Figure 7. Leveling data, topographic relief, and subsurface section simplified from Figure 4. Thesynthetic fit to the data results from spatially varying slip on the Dezghat and Bannh faults (dashed line onsection).

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derived from circuit closure errors in India [Lenox-Conyng-ham, 1916]. In addition, a systematic height-dependent erroris present in the data that is typically less than 1� 10�6 of theheight above the starting point of the line in kilometers. The65 cm of vertical deformation near the Bolan Pass exceedssystematic and random errors in the data by more than anorder of magnitude. The leveling bench marks were geo-referenced from 100 = 1 mile topographic sheets and haveresulting uncertainties of up to 30m (listed byAmbraseys andBilham [2003]). We project these irregularly spaced levelingdata at N110�E along a line perpendicular to the trend offolding in the Kirthar range (Figure 7).[20] No surface rupture was reported for the 1931 Mach

earthquake, although it is possible that West’s [1934] post-seismic investigations did not include traverses across thefrontal thrusts of the Kirthar range except near Sibi. To pro-ceed with the analysis of the leveling data, we assume thatslip on one or more of the mapped subsurface faults wasresponsible for the observed uplift. We digitized these sub-surface faults from the geological cross sections, formingcurved fault segments from a series of contiguous, 3-km-wide, planar segments. We then permitted various combina-tions of contiguous segments to slip. Each segment withnonzero slip contributes to the surface deformation field[Okada, 1992], and we searched for smooth distributions ofslip on contiguous segments that most precisely produced theobserved surface deformation. The analytical procedure thatwe adopted was to shift the surface projection of the param-

eterized faults relative to the leveling data and to invertfor slip using the Green’s functions for each segment. Byminimizing the sum of squared residuals, we determined theoptimal offset between the leveling data and our modeledgeometry.[21] The misfit between the parameterized faults and the

projected leveling data is minimized with slip on deep seg-ments of the Dezgat Thrust with contiguous slip on segmentsof the Bannh fault which branches from it eastward towardthe surface (Figure 7). Maximum observed slip of 3.2 moccurs in segments between 7 and 5 km depth, updip from aninferred interseismic locking line at 8 or 9 km depth, wherewe observe minimum slip. The location of inferred interseis-mic locking is not constrained by observations of interseis-mic deformation, and hence there exists some uncertainty asto its true location; however, in section 6 we undertakeadditional numerical models that are consistent with thisidentification. A deeper locking line would decrease theinferred along strike length of the Mach rupture. The totalwidth of the rupture above our 9-km-deep locking line is42 km and the mean slip is 1.2 m (Table 2). Assuming aseismic moment of 1.1 � 1027 dyn cm, and that all the slipoccurred seismically, the best fit slip distribution requires analong-strike rupture length of 72.2 km.[22] A broader region of uplift near Quetta (Figure 7) can

be explained by invoking minor slip on an uneven decolle-ment or by invoking slip on one of several mapped listricfaults near there, either before or after the 1931 earthquakes.

Table 2. Observed and Synthetic Slip on the Decollementa

Distance (km) Depth (km) Length (km) Dip Obs (m) No Lock (m) Co-8 (m) Co-9 (m) Interseismic (m)

104 �3.5 3.03 7.9 0.28 0.86 0.47 0.463 0101 �3.92 3.01 5.14 0.35 1.34 0.72 0.713 098 �4.19 3.01 3.39 0.3 1.71 0.9 0.901 095 �4.36 3 2.51 0.36 2.03 1.06 1.06 092 �4.5 3 2.47 0.35 2.32 1.21 1.21 089 �4.62 3 3.28 0.4 2.62 1.34 1.35 086 �4.8 3.01 4.96 0.76 2.91 1.48 1.5 083 �5.06 3.03 7.63 1.17 3.2 1.61 1.63 080 �5.46 3.01 4.33 1.21 3.42 1.69 1.73 077 �5.69 3.01 3.83 1.63 3.6 1.73 1.79 074 �5.89 3.01 3.68 2.49 3.75 1.73 1.83 071 �6.08 3.03 7.77 3.16 3.93 1.7 1.84 068 �6.49 3.03 12.79 2.86 4.2 1.59 1.81 065.05 �7.16 3.02 28.75 1.56 4.64 1.31 1.71 062.4 �8.61 3 29.88 0.68 4.96 0 1.37 059.8 �10.11 3.02 12.13 0.54 5.29 0 0 1.656.85 �10.74 3.01 5.68 0.84 5.68 0 0 2.4553.85 �11.04 3.01 4.22 1.04 6.04 0 0 3.1650.85 �11.26 3.01 3.63 0.9 6.35 0 0 3.847.85 �11.45 3 2.38 0.68 6.65 0 0 4.3944.85 �11.58 3 1.96 0.65 6.95 0 0 4.9741.85 �11.68 3 1.71 0.66 7.27 0 0 5.5638.85 �11.77 3 1.49 0.42 7.61 0 0 6.1635.85 �11.85 3 1.29 0.09 7.99 0 0 6.8132.85 �11.91 3 1.11 0.33 8.43 0 0 7.5329.85 �11.97 3 0.96 0.96 8.98 0 0 8.4126.85 �12.02 400.02 0.57 0.48 10 0 0 10

aSegments are free to slip in response to 10 m of thrust displacement imposed on the deepest fault segment, a value scaled to approximate the mean observedcoseismic slip. Co-8 refers to coseismic slip shallower than �8 km depth, and Co-9 refers to coseismic slip from one segment deeper, at �9 km depth. ‘‘NoLock’’ indicates the slip that would occur in the absence of interseismic locking, and Interseismic indicates the synthetic slip that occurs below a locking line at9 km depth. We have scaled the driving element to 10 km so that that synthetic slip approximates the mean slip derived from the observed leveling data.

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We note that the fit to these minor regions of uplift in the dataare nonunique since they are not constrained by well-definedsubsurface geometry.

6. Discussion: The Earthquake Cycle

in a Ramp-Flat-Ramp System

[23] The vertical displacement data fit in section 5 includesall deformation that occurred between 1909 and 1936. Thusthe data include not only coseismic slip but possible post-seismic slip, if any occurred. We show that this is likely witha series of elastic models that assume perfectly frictionlessslip below a locking line, and rupture at shallower depths(Figure 8). The models take the form of those described byFeldl and Bilham [2006] in which a series of contiguous,frictionless boundary elements are driven along a complexrupture surface in an uniform elastic half-space by a far-fielddisplacement imposed at depth. The geometry of this selectedfar-field driving condition is not critical to the models, andsimilar results are obtained by imposing regional contractionon the fault system, or imposed remote thrusting. The bound-ary element computation calculates the amount of sliprequired on contiguous elements to minimize stress in theirvicinity.

[24] The results from three calculations (Table 2 andFigure 8) illustrate models that emulate preseismic, coseismicslip, and finally, slip assuming no interseismic locking belowthe region of coseismic rupture (upper line). For the coseis-mic and preseismic slip calculations, the slip distribution iscalculated by assuming the decollement surface is frictionlessand free to slip in response to 10 m of convergence appliedfrom the west, either for the entire fault surface (no locking)or up to a locking line at 8 or 9 km depth (preseismic). Thecoseismic calculation corresponds to seismic slip above thelocking line, and the input to this model is the static strainfield developed from the preseismic slip distribution deter-mined from the interseismic slip calculation (Table 2).[25] The selection of a 10-m driving displacement is

arbitrary because the synthetic output scales linearly withinput; however, we note that this input value results insynthetic coseismic slip that approximates the coseismic slipinferred from the leveling data. Our 10-m input conditioncorresponds to 2000 years of convergence at 5 mm/yr. Theselection of the locking line, the transition between downdipinterseismic creep and the locked seismogenic rupture zone,was investigated by running models with incrementallyincreased locking depths and by examining the resulting slipdistribution with that inferred from the leveling data. The bestfitting coseismic slip distribution occurs where interseismic

Figure 8. Geometry of the active decollement and frontal thrust (bold line with depth scale on right) andinferred slip on segments shallower than 9 km (gray envelope) compared to synthetic slip (slip scale on left).The calculated slip for the entire fault is given by the top staircase line (twenty-one 3-km-long freelyslipping segments responding to an input displacement of 10 m imposed from the left (west)). The lowerstaircase lines are formed from two calculations: slip anticipated below a locking line at 9 km depth(synthetic preseismic slip) and the slip during rupture at shallower depths that occurs when this interseismicslip distribution drives coseismic rupture (synthetic coseismic). The difference between the two lowerstaircase lines and the upper staircase is the slip deficit caused by interseismic locking at 9 km depth.

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locking occurred above 8 or 9 km depth. The coseismic slipdistributions resulting from locking at each of the 8- and 9-mdepths are listed in Table 2.[26] The spatial distributions of observed and synthetic

coseismic slip show similarities. Peak slip in synthetic andobserved data coincides in updip location but the ratio of peakslip to average slip is less in our numerical experiments thanobserved in the Mach earthquake. Observed slip is twice thesynthetic slip at 6 km depth, and half the synthetic slip at4.5 km depth. No simple changes in fault geometry, or freelyslipping width were able to emulate the localized maximumslip at 6 km depth.[27] The observed minimum in slip that occurs at the

inferred locking line is of special interest. A significant slipdeficit (3 m) occurs here as a result of preseismic and post-seismic pinning at the locking line. The resulting slip deficitis analogous to the slip deficit that occurs between two con-tiguous strike-slip, or normal, faults that slip sequentially.Manighetti et al. [2005] suggests that the stresses generatedby this slip deficit are released in off-fault deformationthrough the creation of secondary faults and folding.[28] A paradox, however, arises in the thrust fault we are

considering, for if the locking line is pinned over severalearthquake cycles, the hanging wall cannot advance over thefootwall. Thus although the minimum slip in the model isconfirmed by the 1909–1936 leveling data, slip may occur atthe locking line at times not sampled by these data. Stressconditions for slip in the region (afterslip) are most favorablefor this translation shortly after the earthquake before theshallow fault ‘‘heals,’’ yet a significant slip deficit remains4 years after the earthquake, thus if slip occurs here it mustdo so over a period of many decades after each earthquake.[29] Alternatively, if the locking line is truly locked for

numerous earthquake cycles, the stored elastic energy theremust eventually be released in a much larger earthquake. Thesignificant variability in slip we see in the Mach earthquakecould in fact be the result of the 1931 event being drivenpartly by elastic energy stored from a previous earthquakecycle. If this occurred, it would suggest that previous earth-quakes terminated at a shallower locking line than that weinfer for 1931. There is some evidence to suggest that theHimalaya may exhibit similar enigmatic behavior, with most

decollement earthquakes associated with 3–7 m of slip andno prominent surface rupture, but with infrequent earth-quakes causing surface ruptures with slip of as much as24 m [Feldl and Bilham, 2006; Bilham and Szeliga, 2008].[30] We discuss next the implied discrepancy between

the amount of convergence (�10 m) required to drive 3 mof coseismic slip of the frontal thrust.

7. Geodetic Convergence, Slip Potential,

and Renewal Time

[31] In many paleoseismic estimates of earthquake recur-rence interval, the renewal time for an earthquake is estimatedfrom the strain rate applied to a fault, a number that is derivedfrom the present-day geodetic displacement rate measured inthe region. Thus one might anticipate that a 3 mm/yr con-vergence rate applied to a fold and thrust belt would permitearthquakes with 3 m of slip every thousand years. Ourperfectly elastic frictionless calculation indicates that therenewal of the Mach earthquake using this approach woulderr by a factor of 8, because only one eighth of the conver-gence is manifest as slip on the frontal thrust (Table 3). Theremaining convergence is presumably accommodated byfolding and thickening of the fold and thrust belt.[32] The effect occurs because the ramp separating the

deep decollement from the shallow frontal thrust acts as abuttress to motion. The ratio of input displacement (geodeticconvergence) to frontal fault slip (measured coseismic slip)depends on the ratio of the depth of the decollement to themean depth of the frontal thrust. As discussed in section 2 thedepth of the decollement may lie at 18–20 km depth. InFigure 8 the depth of the decollement, D, is placed at 12 kmdepth, and the depth of the flat is�6 km, a ratio of�2. From asuite of numerical models in which we varied the decolle-ment depth while maintaining the shallow geometry (Table 3)we derived the following relationship: C = 0.00154D �3.2 m, where C is the (geodetic) convergence of the entirerange andD is the depth of the decollement both measured inmeters. The constants in the equation would be modified insystems with different shallow thrust geometries, but ourbasic finding would be unaltered.[33] We find that where the decollement lies at 6 km, the

renewal time is approximately doubled (Table 3) and whenit lies at 10 km, the renewal time is quadrupled. Only forthe case where the system is a simple planar ramp does therenewal time obey a simple 1:1 relationship between slippotential and geodetic convergence. Note that these calcu-lations are for an infinitely long fault. By reducing the along-strike length of the rupture the ratio of convergence topotential slip is increased yet further, as was found for thesynthetic scaling law for the Himalaya [Feldl and Bilham,2006]. If we use the 18–20 km depth inferred for the depth ofthe decollement underlying the northern Kirthar range, theDezghat thrust ‘‘receives’’ only 12% of the convergenceapplied to the entire Kirthar range between the Chaman faultand the plains of the Indus River.[34] How do we explain this significant discrepancy?

It would appear that the ramp acts as a buttress to sedimen-tary layers driven from the west. Since convergence is not

Table 3. Calculations of Partitioned Convergencea

Depth D (km) Imposed S (m) Ratio D/d Ratio S/s

6 6 1 28 8.9 1.3 310 12 1.7 412 15.75 2 5.218 23.5 2.9 8

aGeometric relations between applied geodetic displacement and slip ontheDezghat/Bannh thrust fault for a range of hypothetical decollement depths(the actual depth is believed to lie in the range 18–20 km). The imposeddisplacement, S, is that calculated to cause the mean observed coseismic slip,s, in the Mach earthquake. D is the mean depth of the decollement, and d isthe approximate starting depth of the frontal thrust above a steeper rampconnecting the two. The ratio S/s is a proxy for the increase in the recurrenceinterval for earthquakes on the frontal thrusts compared to the time that wouldbe calculated from geodetic convergence rates of the entire range.

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released as slip on the shallower fault above and to the east ofthe ramp, it must be manifest as thickening of the sedimentpile to the west. The mean topography west of the ramp is1.5 km higher than to the east of the ramp, and the seismicsection is shortened by�15 km. East of the ramp, the frontalDezgat/Bannh fault system has slipped by <1 km.

8. Sequential Triggering of Ruptures

[35] These three earthquakes are unusual in that most ofthe seismic moment release in the Baluchistan region in thelast 150 years occurred within the 4 years following the firstof these earthquakes (Figure 9). The clustering of these threelarge Baluchistan earthquakes near 30�N has a low proba-bility of occurring by chance; hence some form of triggeringappears probable. The mechanisms of stress transfer arecurrently speculative, especially for the first two events thatoccurred within 66 h of each other. No details of the Sharighearthquakemechanism or its subsurface rupture geometry areknown, and hence static stress changes cannot be computedfor this event. However, the geometric relationship betweenthe Mach and Quetta earthquakes renders a causal linksubstantially easier to comprehend. Rupture of the Dezgatthrust in theMach earthquake reduced compressive east-weststresses and acted, in a sense, to unclamp the fault-normalstress on the Quetta strike-slip fault 60 km to the west. Theinstantaneous Coulomb failure change at Quetta is calculatedto be 10 to 70 kPa depending on the nucleation depth of theQuetta earthquake, a stress change that is more than sufficient

to trigger an earthquake. However, these instantaneous stresschanges were apparently unimportant because the earthquakewas delayed by more than 3 years. If the occurrence of theQuetta earthquake less than 4 years after theMach earthquakeis not a coincidence, we require some form of stress diffusion,or viscous creep.[36] While poroelastic or viscoelastic processes in the

body of the fold and thrust belt, or below it [Freed, 2005]are adequate to cause the observed delay, we consider herean alternative mechanism: viscous creep on the decollementsurface. The rate of propagation of the deformation frontbetween Mach and Quetta, had it occurred linearly, is�18 km/yr. We note that 4 years after the earthquake weinfer afterslip 10 km below the locking line to have amountedto �1 m, with �0.5 m of slip 20 km below the locking line.If this decay rate continued linearly downdip toward theQuetta fault, the slip on the decollement may have amountedto 10–20 cm near Quetta by 1935. The leveling data areinsensitive to slip on a planar, subhorizontal surface, but thebulge in the data east of Quetta suggests that slip of someform occurred 40–50 km west from the Mach event. Slipmay also have occurred prior to the Mach event in the regionbetween Quetta andMach, manifest as vertical changes in the1909–1936 leveling data. We cannot exclude the possibilitythat aseismic or weakly seismic mobility of the structures inthe fold belt prior to the earthquake sequence may have beenresponsible for all three events.

9. Conclusions

[37] Precise leveling data and a fault model derived fromdetailed geological cross sections, permit us to calculate theslip distribution on the rupture surface of the 1931 Machearthquake. We deduce that the earthquake occurred on the42-km-wide (E-W), 72-km-long (N-S) Dezghat/Bannh faultsystem west of Sibi. The fault slipped in a reverse sense up tothe east with maximum slip of 3.2 m and mean slip of 1.2 m.Maximum slip coincides spatially with that predicted inelastic models driven by inferred interseismic stresses, butthe maximum slip is larger than predicted compared to themean slip of the fault. The leveling data suggest that slip alsooccurred downdip of the rupture (�1 m) either as afterslip orslip in other events in the interval 1909–1936.[38] A significant (3–5 m) slip deficit remained near the

interseismic locking line 4 years after the earthquake. Thisslip deficit may now, 67 years later, have been reduced byaseismic processes subsequent to the earthquake, or it mayremain stored as elastic strain to drive future earthquakes.Two mechanisms may act to prevent the accumulation ofseismic deficit over multiple earthquake cycles. The first isthat slow, off-fault deformation, or pressure solution pro-cesses, act to reduce local stresses at the locking line, and thesecond is that infrequent larger earthquakes mine an histor-ically stored slip deficit, accompanied by an incremental shiftin the depth of the locking line. The first is testable inprinciple, in that we could remeasure surviving points ofthe leveling line to determine whether the slip deficit remains.In practice current security issues in the region render thisdifficult. We favor, however, the second mechanism: that

Figure 9. Space-time history of seismic moment releaseas a function of distance from the inferred Mach 1931earthquake rupture zone. More than 89% of the total seismicmoment release in the past 200 years (within a radius of500 km centered on the Mach earthquake) occurred between1931 and 1935. All known earthquakes larger than M6.5 areincluded.

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stored elastic strain from one or more previous earthquakesmay account for the 3.2 m of local slip observed updip fromthe inferred locking line in the 1931 earthquake. The localmaximum slip is easier to explain as an additional 1.5–2m ofslip inherited from strain unreleased by former earthquakes,than the alternative solution, that the slip in the earthquakeshould everywhere have been �3 m in 1931 and that an �2-m slip deficit remains on most of the fault.[39] In considering the details of the Mach earthquake we

examined in numerical experiments the elastic processesprevailing during the entire seismic cycle. We found that slipon a frontal thrust is always less than the geodetic contractionrate of a fold and thrust belt, unless the frontal thrust consistsof a planar fault. In the case of a geometrically complexunderlying thrust fault with variable dip, we find that parti-tioning of slip to the frontal thrusts is reduced in proportion tothe ratio of decollement depth to shallow thrust depth wherethese are separated by a ramp. This significant discrepancybetween the geodetic loading rate and the slip potential offrontal faults of the fold belt is presumably responsible forthickening the pile of sediments by folding and listric fault-ing. Partitioning in the Kirthar range, as elsewhere, results ina significantly longer renewal time for earthquakes on thefrontal fault of the range, than would be derived from thegeodetic convergence rate alone. Thus, although the weaklyconstrained 5 mm/yr GPS convergence rate between Quettaand Sibi would result in a minimum renewal time for 1.2 m(average slip) on a planar frontal thrust fault of 240 years, ourstudy suggests that the presence of a decollement at �18 kmdepth would extend the recurrence interval for Mach-type

earthquakes on the Dezghat/Bannh fault system by a factor of8, to �2000 years.[40] Given that more than 90% of the seismic moment

release in the region occurred between 1930 and 1935 webelieve that sequential triggering of the three earthquakesoccurred. The 1931 Sharigh earthquake was clearly respon-sible for triggering the Mach earthquake 3 days later but thestructural relationship between these two fault systems isobscure, and hence any attempt to explain the mechanismsinvolved are at present speculative. In contrast, the ‘‘broad-side’’ relation between the Mach and Quetta rupture zones isconsistent with an increase in Coulomb failure stress on theQuetta fault at the time of the Mach earthquake. The 3.5-yearinterval between the two earthquakes indicates, however, thatstatic stresses changes alone were insufficient to trigger theQuetta earthquake.We hypothesize that theMach earthquakereduced east-west stresses on the decollement/ramp systemthat would have facilitated accelerated creep on the basaldecollement beneath the Kirthar range. Deformation betweenMach and Quetta inferred from minor uplift and subsidencein the leveling data, are consistent with strain changesaccompanying decollement slip, although interpretation ofthese data are nonunique.

[41] Acknowledgments. For numerous interesting discussions we areindebted to Bob Anderson, Peter Molnar, Karl Mueller, and others. Wethank Andrew Meigs and an anonymous reviewer for their thoroughreviews. Miriam Garcia received a UNAVCO internship and R.B. thanksGeof King and Paul Tapponier of L’institute de Physique du Globe for theirhospitality during the writing of his paper. The study was funded by theNational Science Foundation grants EAR003449 and EAR 0739081.

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